Are there any clever tricks for the data processing involved here, given that the delay is a shift in the time domain and the Doppler effect is a shift in the frequency domain? Maybe involving fractional Fourier transforms, or wavelets?

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Lots of clever tricks, all ITAR controlled.

Uninformed FUD, not a single dsp "trick" related to passive radar is ITAR controlled. The equivalence the OP mentioned is literally described in every undergraduate dsp textbook.

k